Calculus i tunc geveci

Calculus I covers the usual topics of the rst semester: Limits, continuity, the ative, the integral and special functions such exponential functions, logarithms, semester: Further techni

Tunc Geveci

Calculus I: First Edition

]

reproduced, transmitted, or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information retrieval system without the written permission of University Readers, Inc.

First published in the United States of America in 2011 by Cognella, a division of University Readers, Inc.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.

15 14 13 12 11 1 2 3 4 5

Printed in the United States of America

ISBN: 978-1-935551-42-3

1.1 Powers of x, Sine and Cosine 1

1.2 Combinations of Functions 16

1.3 Limits and Continuity: The Concepts 31

1.4 The Precise Denitions (Optional) 41

1.5 The Calculation of Limits 47

1.6 Innite Limits 58

1.7 Limits at Innity 68

1.8 The Limit of a Sequence 79

2 The Derivative 93 2.1 The Concept of the Derivative 93

2.2 The Derivatives of Powers and Linear Combinations 107

2.3 The Derivatives of Sine and Cosine 121

2.4 Velocity and Acceleration 131

2.5 Local Linear Approximations and the Dierential 138

2.6 The Product Rule and the Quotient Rule 148

2.7 The Chain Rule 157

2.8 Related Rate Problems 167

2.9 Newton’s Method 174

2.10 Implicit Dierentiation 184

3 Maxima and Minima 193 3.1 Increasing/decreasing Behavior and Extrema 193

3.2 The Mean Value Theorem 205

3.3 Concavity and Extrema 214

3.4 Sketching the Graph of a Function 226

3.5 Applications of Maxima and Minima 233

4 Special Functions 249 4.1 Inverse Functions 249

4.2 The Derivative of an Inverse Function 262

4.3 The Natural Exponential and Logarithm 272

4.4 Arbitrary Bases 285

4.5 Orders of Magnitude 291

4.6 Exponential Growth and Decay 302

4.7 Hyperbolic and Inverse Hyperbolic Functions 318

4.8 L’Hôpital’s Rule 332

iii

5 The Integral 347

5.1 The Approximation of Area 347

5.2 The Denition of the Integral 357

5.3 The Fundamental Theorem of Calculus: Part 1 371

5.4 The Fundamental Theorem of Calculus: Part 2 385

5.5 Integration is a Linear Operation 403

5.6 The Substitution Rule 415

5.7 The Dierential Equation y0= f 427

A Precalculus Review 435 A.1 Solutions of Polynomial Equations 435

A.2 The Binomial Theorem 440

A.3 The Number Line 442

A.4 Decimal Approximations 451

A.5 The Coordinate Plane 457

A.6 Special Angles and Trigonometric Identities 471

This is the rst volume of my calculus series, Calculus I, Calculus II and Calculus III Thisseries is designed for the usual three semester calculus sequence that the majority of science andengineering majors in the United States are required to take Some majors may be required totake only the rst two parts of the sequence

Calculus I covers the usual topics of the rst semester: Limits, continuity, the ative, the integral and special functions such exponential functions, logarithms,

semester: Further techniques and applications of the integral, improper integrals,linear and separable rst-order dierential equations, innite series, parametrizedcurves and polar coordinates Calculus III covers topics in multivariable calculus:Vectors, vector-valued functions, directional derivatives, local linear approxima-tions, multiple integrals, line integrals, surface integrals, and the theorems of Green,Gauss and Stokes

An important feature of my book is its focus on the fundamental concepts, essentialfunctions and formulas of calculus Students should not lose sight of the basic conceptsand tools of calculus by being bombarded with functions and dierentiation or antidierentia-tion formulas that are not signicant I have written the examples and designed the exercisesaccordingly I believe that "less is more" That approach enables one to demonstrate to thestudents the beauty and utility of calculus, without cluttering it with ugly expressions Anotherimportant feature of my book is the use of visualization as an integral part of the expo-sition I believe that the most signicant contribution of technology to the teaching of a basiccourse such as calculus has been the eortless production of graphics of good quality Numericalexperiments are also helpful in explaining the basic ideas of calculus, and I have included suchdata

Appendix A and the rst two sections of Chapter 1 provide a review of the essential precalculusmaterial that the student should know in order to meet the challenges of calculus The studentshould be comfortable with the language and notation that are necessary in order to refer tofunctions unambiguously That is why I have included such material in the beginning of the rstchapter The main goal of that chapter is to introduce the concepts of limits and continuity,and to provide the student with the necessary tools that are helpful in the calculation of limits

I nd it practical to introduce the concepts of limits and continuity simultaneously in terms ofthe understanding of the concepts and the evaluation of limits Thus, my treatment diers fromthe usual calculus text I also treat innite limits and limits at innity with more care than theusual calculus text, and provide the student with more tools for the evaluation of such limits.The limit of a sequence is treated in this chapter since the language of sequences is convenient inthe discussion of Newton’s method and the convergence of certain Riemann sums to the integral.The usual calculus text postpones the discussion of sequences to the chapter on innite series.Chapter 2 introduces the derivative I deviate from the usual calculus text by discussing locallinear approximations and the dierential at an early stage Indeed, the idea of derivative is

v

intimately linked to local linear approximations with a certain form of the error, and that is howthe concept is generalized to functions of several variables At this point, the early discussion

of local linear approximations is helpful in justifying the identication of the derivative of afunction with its rate of change and the slope of its graph Such discussion is also helpful inproviding plausibility arguments for the product rule and the chain rule

Chapter 3 discusses the link between the sign of the derivative, the increasing/decreasing havior of a function, and its local and global extrema Unlike the usual text, I do not start

be-by the stating the theorem on the existence of the absolute extrema of a continuous function

on a closed and bounded interval and the Mean Value Theorem Thus, my approach is morepractical, and does not give the impression to the student that the only time you can talk aboutabsolute extrema is when you have a continuous function on a closed and bounded interval Ialso discuss the link between the second derivative, the increasing/decreasing behavior of the

rst derivative, and the concavity of a graph

Chapter 4 introduces special functions such as exponential and logarithmic functions and inversetrigonometric functions The introduction of inverse functions in the usual text is confusing Myintroduction is more practical and more careful at the same time I postpone the introduction ofthe exponential and logarithmic functions to this chapter since I nd it impossible to motivatethe signicance of the natural exponential function before introducing the derivative Besides,powers, sine, cosine, and their combinations are adequate for the illustration of the derivativeand its applications prior to this chapter I discuss the dierent orders of magnitude of powers,exponential and logarithms independently of L’Hôpital’s rule I nd this approach much moreilluminating than a mechanical applications of L’Hôpital’s rule (that is covered in the last section

of the chapter)

Chapter 5 introduces the integral I introduce the part of the Fundamental Theorem of Calculus

a

F0(x) dx = F (b)  F (a)

rst, since that enables the student to compute many integrals before the introduction of theidea of a function that is dened via an integral Thus, the student has a better chance ofunderstanding the meaning of the part of the Fundamental Theorem which says that

ddx

Z x

a f(t)dt = f(x)(provided that f is continuous) Many texts introduce both parts of the Theorem suddenly, and

do not present them in a way that establishes the link between the derivative and the integralclearly I nd it amusing, but not helpful, when I see a title such as "total change theorem", as

if something other than the Fundamental Theorem is involved

My own preference is to cover special functions before the integral since that makes it possible

to provide a richer collection of examples and problems in the following chapter On the otherhand, some people prefer the elegance of the "late transcendentals" approach whereby thenatural logarithm is introduced as an integral The pdf les for the versions of chapter 4 andchapters 5 that introduce the integral before logarithms, exponentials and inverse trigonometricfunctions will be provided upon request

Remarks on some icons: I have indicated the end of a proof by¥, the end of an example by

¤ and the end of a remark by 

Supplements: An instructors’ solution manual that contains the solutions of all the lems is available as a PDF le that can be sent to an instructor who has adopted the book Thestudent who purchases the book can access the students’ solutions manual that contains thesolutions of odd numbered problems via www.cognella.com

prob-PREFACE vii

Acknowledgments: ScienticWorkPlace enabled me to type the text and the mathematicalformulas easily in a seamless manner Adobe Acrobat Pro has enabled me to convert theLaTeX les to pdf les Mathematica has enabled me to import high quality graphics to mydocuments I am grateful to the producers and marketers of such software without which Iwould not have had he patience to write and rewrite the material in this volume

I would also like to acknowledge my gratitude to two wonderful mathematicians who haveinuenced me most by demonstrating the beauty of Mathematics and teaching me to writeclearly and precisely: Errett Bishop and Stefan Warschawski

Last, but not the least, I am grateful to Simla for her encouragement and patience while I spenthours in front a computer screen

Tunc Geveci (tgeveci@math.sdsu.edu)

San Diego, March 2010

Functions, Limits and Continuity

The rst two sections of this chapter review some basic facts about functions dened by nal powers of x, polynomials, rational functions and trigonometric functions Appen-dix A contains additional precalculus review material We will discuss exponential, logarithmicand inverse trigonometric functions in Chapter 4

ratio-The main body of the chapter is devoted to the discussion of the fundamental concepts ofcontinuity and limits Roughly speaking, a function f is said to be continuous at a point a if

f(x) approximates f (a) when x is close to a It may happen that f (x) approximates a specicnumber L if x is close to a but x6= a, even if f is not dened at a, or irrespective of the value

of f at a The relevant concept is the limit of f at a We will also discuss innite limits,limits at innity and the limits of sequences

We will deal with a variety of functions in calculus We will begin by reviewing the relevantterminology and notation Then we will review some basic facts about sin (x), cos (x) andrational powers of x These are the building blocks for a rich collection of functions

Terminology and Notation

Denition 1 Let D be a subset of the set of real numbersR A real-valued functionof a realvariable with domain D is a rule that assigns to each element of D a unique real number

We may refer to a function by a letter such as f or g Some functions are special since theyoccur frequently Such functions have names such as sine or cosine, and their names have specicabbreviations, such as sin or cos We will denote an arbitrary element of the domain of a function

by a letter such as x or t If we choose the letter x, and the function in question is f , then x isthe independent variable of f The unique real number that is assigned by f to x is thevalue of f at x and is denoted by f (x) (read “f of x”) The value of a special function will bedenoted by using the abbreviation reserved for that function For example, the value of the sinefunction at x will be denoted by sin(x) If we set y = f (x), then y is the dependent variable

of f We can refer to a function f by the letter that denotes the dependent variable and set

2 CHAPTER 1 FUNCTIONS, LIMITS AND CONTINUITY

We can express the domain of f as (, 0)  (0, +), the union of the interval (, 0) and(0, +) (Section A3 of Appendix A contains a review of the number line and intervals) Theindependent variable is x If we set y = 1/x, the dependent variable of f is y We can replace

x by any nonzero real number to obtain the corresponding value of the function For example,the value of f at

2 is

f(2) = 1

2 = 0.707107,rounded to 6 signicant digits (we count the number of signicant digits of a decimal startingwith the rst nonzero digit, as discussed in Section A4 of Appendix A) Thus, the value of thedependent variable y that corresponds to the value

2 of the independent variable is 1/

2 ¤Example 2 Assume that a car is traveling at a constant speed of 60 miles per hour If wedenote time by t (in hours), the distance s covered by the car in t hours is 60t miles Let usset s = f (t) = 60t The letter t denotes the independent variable and can be assigned anynonnegative real number Thus, the domain of f is the set of all nonnegative real numbers andcan be expressed as the interval [0,) The letter s denotes a dependent variable ¤

Example 3 The surface area A of a sphere of radius r is 4r2 Let us set A = g (r) = 4r2for any r > 0 The domain of the function g is the interval (0, +) The independent variable

is r and the dependent variable is A We may choose to refer to the function by the letter thatdenotes the dependent variable, and set A(r) = 4r2 ¤

Eventually, we will consider relationships between variable entities that need not be real bers For example, the variable in question can be a point whose coordinates in the Cartesiancoordinate plane are real-valued functions of a real variable We will speak of real-valuedfunctions of a real variable simply as functions, until we consider more general relation-ships

num-Assume that f (x) is a single expression for each x The natural domain of f consists of all

x R such that the expression f(x) is a real number We may refer to “the function f(x)” Inthis case, it should be understood that the domain of the function f is its natural domain Forexample, if f (x) = 1/x, the natural domain of f consists of all nonzero real numbers We mayrefer to f as “the function 1/x”

The graph of a function is very helpful in visualizing the relationship between the dependentand independent variable If x denotes the independent variable of the function f and y denotesthe dependent variable, the graph of f is the subset of the xy-plane that consists of the points(x, y) where x is in the domain of f and y = f(x):

¶: x 6= 0

¾

Since 1/x attains values of arbitrarily large magnitude if x is near 0, a graphing utility shows

us only part of the graph of f corresponding to an interval that contains 0 We will say thatthe viewing window is[a, b] × [c, d] if x and y are restricted to the intervals [a, b] and [c, d],respectively Figure 1 shows the part of the graph of f in the viewing window [3, 3] × [5, 5]

We will usually refer to such a picture simply as the graph of the relevant function ¤

10

5

5 10 y

Figure 1

You are already familiar with the graphs of certain types of functions from precalculus courses.You should be able to provide rough sketches of functions that are not too complicated Infact, calculus will provide you with certain tools that will enable you to come up with sketchesthat reect the behavior of many functions correctly A graphing calculator or a softwarepackage such as Maple, Mathematica, Matlab or MuPAD is helpful in obtaining the graph

a function We will refer to such a device as a graphing utility if we wish to emphasize itsgraphing capabilities, and as a computational utility if we wish to emphasize its computa-tional capabilities If a particular example or exercise requires both computational and graphicalcapabilities, we may simply refer generically to “a calculator” Maple, Mathematica and Mu-PAD are computer algebra systems, i.e., they can work with symbols such as x and y, aswell as with numbers (Matlab has a “symbolic toolbox” that is powered by Maple) We will usethe abbreviation “CAS” when we refer to computer algebra systems

Figure 1 was generated with the help of a graphing utility Note that the scale on the verticalaxis is not the same as the scale on the horizontal axis That will be the case in many of thegraphs that will be displayed in this book, since a graphing utility adjusts the scales on thecoordinate axes automatically for better viewing We will impose equal scales on the axes ifthat serves a denite purpose Also note that we have labeled both axes in Figure 1 We maynot label the vertical axis in some cases where we do not assign a letter to the dependent variable

of the relevant function

Figure 2

4 CHAPTER 1 FUNCTIONS, LIMITS AND CONTINUITY

In Section A5 of Appendix A we review the graphs of some equations in the xy-plane If f (x)

is dened by the same equation for each x in its domain, the graph of f in the xy-plane is thesame as the graph of the equation y = f (x) On the other hand, it is not true that the graph of

an equation is always the graph of a function The vertical line test is helpful in determiningwhether a graph is the graph of a function:

Assume that f is a function Given a vertical line x = a, either the graph of f has

no point on the line, or there is a single point where the graph of f and the line

x = a intersect

Indeed, if a is not in the domain of f then f does not assign a value to a, so that the graph

of f does not have any point on the line x = a If a is in the domain of f then f assigns theunique value f (a) to a, so that the only point on the line x = a that belongs to the graph of f

2

2 y

Figure 4

In Figure 4 it appears as if the vertical line segment that connects the points (1, 1) and (1, 2) ispart of the graph of f , and that is not the case We will refer to such a line as a spurious linesegment The only point on that line segment that belongs to the graph of f is the point (1, 2).

We can explain the picture as follows: The graphing utility that produced Figure 4 sampled avalue of x slightly to the left of 1, another value of x slightly to the right of 1, and connected thecorresponding points by a line segment One point is very close to (1, 1) and the other point isvery close to (1, 2) The spurious line segment is the line segment that joins these points Figure

2 was produced by a graphing utility that makes special provision for the sudden jump in thevalue of the function at x = 1 Some pictures in this book may contain spurious line segments,just as the pictures that are produced by your graphing calculator As long as we interpret thepictures correctly, there should be no misunderstanding 

We should be clear about the meaning of the equality of functions: We say that the function

Example 8 Determine whether f = g if

a) f (r) = 4r2for any r R, and g(x) = 4x2 for any x R,

b)f (x) = 4x2 for any x R and g(x) = 4x2 for any x 0

Solution

a) The domain of g is the same as the domain of f and consists of all real numbers Since

g(x) = 4x2= f(x)for each x  R, we have g = f, even though we have used dierent letters to denote theindependent variables

b) Even though g(x) = f (x) for each x 0, the function g is not equal to the function f, sincethe domain of g is the set of nonnegative real numbers [0, +), whereas the domain of f is theset of all real numbers R The graph of g is part of the graph of f , as shown in Figure 5 ¤

40 80 y

y  fx

40 80 y

y  gx

Figure 5

Example 8 illustrates the restriction of a function: The function g is the restriction of thefunction f to the set S if the domain of g is S, the set S is contained in the domain of f ,and g(x) = f (x) for each x S In this case, f is said to be an extension of g Thus, if f and

g are as in part b) of Example 8, then g is the restriction of f to [0, +) The function f is anextension of g

We have many uses for the absolute value, as reviewed in Section A3 of Appendix A Let uslook at the relevant function:

6 CHAPTER 1 FUNCTIONS, LIMITS AND CONTINUITYExample 9 (The absolute-value function) Let

Figure 6: The absolute-value function

Functions dened by rational powers of x are building blocks for a rich family of functions, as

we will see throughout calculus Let’s review some basic facts about such functions and look atsome samples

Let f (x) = xu, where the exponent r is a rational number The rules for exponents should

be familiar If a and b are rational numbers, we have

xaxb = xa+b,

xa,(xa)b = xab,provided that the expressions are dened

If f (x) = xn, where n is a positive integer, f (x) is dened for each x  R, so that we canidentify the natural domain of f with the entire number line

If f (x) = x, then f is a linear function The graph of f is a line with slope 1 that passes throughthe origin Since the value of f at x is the same as x, f can be referred to as the identityfunction

y  x

Figure 7: The identity function

If f (x) = x2, then f is a quadratic function, and the graph of f is a parabola The function is

a prototype of the functions dened by xn, where n is an even positive integer Figure 8 showsthe graphs of y = x2 and y = x4

1 2 3 4

yx3

yx5

Figure 9

A function f is an even function if f (x) = f (x) for each x in the domain of f A function f

is an odd function if f (x) = f (x) for each x in the domain of f The graph of an evenfunction is symmetric with respect to the vertical axis, and the graph of an oddfunction is symmetric with respect to the origin Indeed, if f is even and (x, y) is on thegraph of f , we have y = f (x) = f (x) Therefore, (x, y) is also on the graph of f, and thepoints (x, y) and (x, y) are symmetric with respect to the vertical axis If f is odd and (x, y)

is on the graph of f , then y = f (x) =f (x), so that y = f (x) Therefore, (x, y) isalso on the graph of f , and the points (x, y) and (x, y) are symmetric with respect to theorigin

If f (x) = xn, where n is an even positive integer, then f (x) = (x)n = xn = f (x), so that

f is an even function Therefore, the graph of f is symmetric with respect to the vertical axis.Figure 8 that shows the graphs of y = x2 and y = x4is consistent with that fact

If f (x) = xn, where n is an odd positive integer, then f (x) = (x)n= xn= f (x), so that

f is an odd function Therefore, the graph of f is symmetric with respect to the origin Figure

9 that shows the graphs of y = x3 and y = x5 is consistent with that fact

If n is a positive integer, and

f(x) = xn= 1

xn,the natural domain of f consists of all real numbers x such that x6= 0 We can identify thedomain of f with the union of the open intervals (, 0) and (0, +) Such a function is odd

if n is odd and even if n is even

Figure 10 shows the graph of f , where f (x) = x1= 1/x

8 CHAPTER 1 FUNCTIONS, LIMITS AND CONTINUITY

20

10

10 20 y

Figure 10

Note that the graph of f is symmetric with respect to the origin, consistent with the fact that

f is an odd function Also note that|1/x| becomes arbitrarily large if we imagine that x takes

on values that are closer and closer to 0 The function f is a prototype of functions dened by

xn, where n is an odd positive integer

Figure 11 shows the graph of g, where g (x) = x2= 1/x2

50

100 y

Figure 11

Note that the graph of g is symmetric with respect to the vertical axis, consistent with the factthat g is an even function Also note that 1/x2becomes arbitrarily large if we imagine that xtakes on values that are closer and closer to 0.The function g is a prototype of functions dened

by xn, where n is an even positive integer

If n is an even positive integer and x 0, we set

y = x1@q=q

x if x = yq and y 0

Thus, the natural domain of a function dened by x1/n is the interval [0, +) if n is an evenpositive integer

The square-root function dened by x1/2is a prototype of functions dened by x1/n, where

n is an even positive integer:

y

y x12

y x14

Figure 12

If n is an odd positive integer, we set

y = x1@q=q

x if x = yq

Here, x can be an arbitrary real number Thus, the natural domain of a function dened by

x1/n, where n is an odd positive integer, is the entire number line

The cube-root function dened by x1/3is a prototype of functions dened by x1/n, where n

is an odd positive integer:

y= x1/3=3

x if x = y3.Figure 13 shows the graphs of y = x1/3and y = x1/5 Note that the graphs are symmetric withrespect to the origin, since the underlying functions are odd (conrm)

In general, graphing utilities do not picture the part of the graph of f near the vertical axis verywell The picture may give the impression that part of the graph coincides with the verticalaxis This is not the case, of course The function is not dened at 0 In fact, f (x) takes onarbitrarily large values if x is close to 0

2 4 6

For example, if we set

f(x) = x2/3=³

x1/3

´2,

10 CHAPTER 1 FUNCTIONS, LIMITS AND CONTINUITYthen f (x) is dened for each x R The function is even since

Figure 15

Sine and Cosine

Let’s begin by reviewing the meaning of radian measure Consider the circle that is centered

at the origin and has radius 1 We will refer to this circle as the unit circle The unit circle isthe graph of the equation u2+ v2= 1 in the uv-plane We associate a unique point on the unitcircle with each real number x as follows: If x = 0, the associated point is A = (1, 0) Imagine

a particle that travels on the unit circle, starting at the point A If x > 0, we associate with

x the point P that the particle reaches by traveling the distance x along the unit circle in thecounterclockwise direction With reference to Figure 16, x is the radian measure of the angleAOP Thus, the radian measure of the angle AOP is the length of the arc AP

1

1 v

A O

P x

Figure 16

If x < 0, imagine that the particle travels the distance |x| = x along the unit circle in theclockwise direction and reaches the point P We associate the point P with the number x Withreference to Figure 17, x is the radian measure of the angle AOP

1 u

1

1 v

A O

P

x

Figure 17

The length of the unit circle is 2 Thus, the point that is associated with x + 2n, where

n= 0, ±1, ±2, , is the same as the point that is associated with x

In everyday usage, we refer to angles in degrees Since one revolution around the unit circlecorresponds to 360o, if the radian measure of an angle is x and the degree measure of an angle

is o, we have the relationship

In particular, 180o corresponds to  radians, 90o corresponds to /2 radians, 60o corresponds

to /3 radians, 45o corresponds to /4 radians, and 30o corresponds to /6 radians In thisbook, “the default measure” for angles will be radians, so that if we refer to “theangle x”, it should be understood that x is in radians, unless stated otherwise Therewill be ample opportunity for you to appreciate the fact that the radian measure is the naturalmeasure for angles in calculus

The calculation of the coordinates of the point on the unit circle that is associated with a realnumber x leads to the trigonometric functions sine and cosine:

Denition 2 If P = (u, v) is the point on the unit circle that is associated with the real number

x, we dene the horizontal coordinate u to be the value of the function cosine at x, and thevertical coordinate v as the value of function sine at x We abbreviate sine as sin and cosine ascos, so that P = (cos (x) , sin(x))

1

1 v

P

x cosx

sinx

Figure 18: The denition of sin (x) and cos (x)

The notations cos(x) and sin(x) are consistent with notation f (x) that denotes the value of afunction f at x Traditionally, cos x and sin x denote cos (x) and sin (x), respectively Thetraditional imprecise notation may lead to confusion in describing functions such as sin¡

x2¢.Besides, a calculator will demand precise syntax Therefore, the notations cos (x) and sin(x)will be used in this book, and you are encouraged to do the same

12 CHAPTER 1 FUNCTIONS, LIMITS AND CONTINUITY

We have

1 sin (x) 1 and  1 cos (x) 1for each real number x This follows from the geometric denitions of the functions: If P =(cos (x) , sin (x)), the horizontal and vertical coordinates of the point P are between 1 and 1,since P is on the unit circle

The functions sine and cosine are periodic functions First, let us make sure that we agree

on the meaning of periodicity:

Denition 3 A function f is said to be periodic with period p (p6= 0) if f(x + p) = f(x)for each real number x

If f is periodic with period p, we have f (x± np) = f(x + p) for n = 1, 2, 3, , so that ±np isalso a period of f The smallest positive period of f is referred to as its fundamental period.The functions sine and cosine are periodic with period2 Indeed, let P = (cos(x), sin(x))

be the point on the unit circle that is associated with the real number x The point P is ciated with x + 2, as well, since the particle that travels on the unit circle comes back to thesame point after a full revolution around the unit circle Therefore,

asso-(cos (x + 2) , sin (x + 2)) = asso-(cos (x) , sin (x)) ,

so that cos (x + 2) = cos (x) and sin (x + 2) = sin (x)

Since sine and cosine are periodic with period 2, the shape of the graph of each function on aninterval of the form [4, 2], [2, 0] or [2, 4] is the same as the shape of the graph of thefunction on [0, 2] We may choose any interval of length 2 to discuss an issue pertaining tosine or cosine The most popular choices are the intervals [0, 2] and [, ] Figure 19 showsthe graphs of sine and cosine on the interval [2, 2]

1

1 y

y  sinx

2 Π Π Π 2 Πx

1

1 y

y  cosx

Π 2

Figure 9

Figure 19 indicates that the graph of sine is symmetric with respect to the origin, and the graph

of cosine is symmetric with respect to the vertical axis These graphical observations reect

“the symmetry” of each function:

Sine is an odd function and cosine is an even function, i.e.,

sin(x) =  sin(x) and cos(x) = cos(x)for each real number x

Figure 20 illustrates these facts for x between 0 and /2.

1

1 v

The functions sine and cosine are “built-in” functions in any calculator that you may be using

in conjunction with calculus, in the sense that the calculator is equipped with the capability tocompute sin(x) and cos(x) accurately and fast A computer algebra system can compute theexact values of sin(x) and cos(x) if x is a special angle such as /3 or /6 In general, a calculatorwill provide you with approximations to sin(x) and cos(x) Your own accuracy requirements,and the capability of your calculator may vary As discussed in Section A4 of Appendix A, therelevant numbers in this book have been calculated with the help of a calculator that bases itscalculations on rounding decimals to 14 signicant digits

Example 10 Determine the points on the unit circle that are associated with ±/6 Sketchthe unit circle and indicate the location of each point on the unit circle

Solution

As discussed in Section A6 of Appendix A,

cos³6

´

=

3

³6

´

2.Therefore, the point P that is associated with /6 is

³cos³6

´,sin³6

!

Since cosine is an even function and sine is an odd function, we have

³

6´=  sin³6´= 12.Therefore, point Q that is associated with/6 is

³cos³

The points P and Q are shown in Figure 21 ¤

14 CHAPTER 1 FUNCTIONS, LIMITS AND CONTINUITY

1 Let V be the volume of a spherical balloon of radius r Express V as a function of r What

is the volume of such a balloon of radius 40 cm.?

2 Let V be the volume of a right circular cylinder of height 20 inches and base radius r inches.Express V as a function r What is the volume of such a cylinder of base radius 5 inches?

3 Let A be the area of the lateral surface of a right circular cylinder of base radius 50 cm andheight h cm Express A as a function of h What is the lateral area of such a cylinder of height

f(x) = 4 + x2

9  x211

f(x) = 1 + x + x2+ x3

(x2+ 1)2(x2 1)12

f(x) =

x 313

x3/416

f(x) = (4  x2)2/317

f(x) = (x2 9)3/418

f(x) = sin(x) +1

3sin(3x) +

1

5sin(5x)

In problems 19 - 22, show that the graph of the given equation is not the graph of a function of

x Sketch the graphs

a) Sketch the graphs of f and g

b) Show that g is an increasing function Is your sketch consistent with this fact?

26 Let f (x) = x2+ 6x + 13 for each x  R and let g be the restriction of f to the interval(, 3]

a) Sketch the graphs of f and g

b) Show that g is a decreasing function Is your sketch consistent with this fact?

27 [C] Make use of your graphing utility to plot the graphs of the following functions for x[1, 1] Are the pictures consistent with the fact that the functions are odd?

a) x5

b) 1

x3

28 [C] Make use of your graphing utility to plot the graphs of the following functions for

x [1, 1] Are the pictures consistent with the fact that the functions are even?

a) x6

16 CHAPTER 1 FUNCTIONS, LIMITS AND CONTINUITY

b) 1

x4

29 Let g be the restriction of the sine function to the interval [/2, /2]

a) Sketch the graph of g Is your sketch consistent with the fact that g is an increasing function?b) Is g an odd function or an even function? Is your sketch consistent with your response?

30 Let g be the restriction of the cosine function to the interval [0, ]

a) Sketch the graph of g Is your sketch consistent with the fact that g is a decreasing function?b) Does it make sense to enquire whether g is an even or odd function?

31 Sketch the point on the unit circle that correspond to x (radians) Determine cos (x) andsin (x) Is the picture consistent with the fact that sine is an odd function and cosine is an evenfunction?

a) x = /3 and x =/3

b) x = 5/6 and x =5/6

32 [C] Make use of your computational utility to calculate sin (x) and cos (x) (x is in the defaultmode of radians) Round decimals to 6 signicant digits, as discussed in Appendix A4 Are thenumbers consistent with the fact that sine is an odd function and cosine is an even function?a) x = 1 and x =1

b) x = 2.5 and x =2.5

33 [C] Make use of your computational utility to calculate sin (x) and cos (x) (x is in the defaultmode of radians) Round decimals to 6 signicant digits Are the numbers consistent with thefact that sine and cosine are periodic functions with period 2?

of functions

Arithmetic Combinations of Functions

Let f and g denote arbitrary functions Recall that the sum, product and quotient of f and

g are dened via the corresponding operations on their values: For each x that belongs to thedomain of f and the domain of g,

(f + g) (x) = f (x) + g (x) ,(fg) (x) = f (x) g (x) ,

fg

¶(x) = f(x)

g(x)provided that g (x)6= 0

Example 1 Let f (x) = x and g (x) = cos (x) Then

(f + g) (x) = f (x) + g (x) = x + cos (x)for each x R

Figure 1 shows the graph of f + g on the interval [2, 2] The dashed line in Figure 1 is theline y = f (x) = x Note that the point (x + cos(x), x) on the graph of f + g can be obtained bymoving the point (x, x) on the graph of f vertically by g (x) = cos(x) (up or down, depending

on the sign of cos(x)) ¤

6

4

2 2 4 6 y

x and sin (x) are dened, i.e., for each x  0 Thus, the domain of theproduct f g is the interval [0, +) Since 1 sin (x) 1, we have

x xsin (x) x for each x R

Therefore, the graph of f g lies between the graphs of y =

18 CHAPTER 1 FUNCTIONS, LIMITS AND CONTINUITY

x and g (x) =

x+ 2 Then g (x) = f (x) + 2 Figure 3 shows thegraphs of f and g If we imagine that the graph of f is shifted vertically upward by by 4 units,the resulting curve will coincide with the graph of g

2 4 6 y

2

2 4 y

We will use special terminology to refer to the products of constant functions with arbitraryfunctions: If c is a constant, the constant multiple cf of the function f is dened so that

(cf) (x) = cf (x)for each x in the domain of f

Example 4 Let f (x) = x2 Then

(2f) (x) = 2f (x) = 2x2,

1

2f

¶(x) =1

Figure 5 displays the graphs f , 2f ,2f, (1/2) f and (1/2) f The graph of 2f can be sketched

by stretching the graph of f vertically by a factor of 2 The graph of f /2 can be sketched

by shrinking the graph of f vertically by a factor of 2 The graph of2f can be sketched byreecting the graph of 2f with respect to the horizontal axis The graph off/2 can sketched

by reecting the graph of f /2 with respect to the horizontal axis ¤

4

4 y

Many new functions are formed by adding constant multiples of basic functions:

Denition 1 A linear combination of the functions f and g is a function of the form

c f + c2g, where c1and c2are constants

Thus,

(c1f+ c2g) (x) = c1f(x) + c2g(x)for each x such that both f (x) and g (x) are dened

We can form linear combinations of more than two functions in the obvious manner Forexample, if c1, c2and c3are constants, and f1, f2, f3are given functions,

(c1f1+ c2f2+ c3f3) (x) = c1f1(x) + c2f2(x) + c3f3(x)for each x such that f1(x), f2(x) and f3(x) are dened

A word of caution: You must distinguish between the meaning of “linear” as in a “linearfunction”, and “linear” as in “a linear combination” For example, the quadratic function

1 + x + x2is a linear combination of 1, x and x2, but the function is not linear

Note that a linear function f (x) = mx + b is a linear combination of the constant function 1and x A quadratic function ax2+ bx + c is a linear combination of 1, x and x2 More generally,

a polynomial is a linear combination of 1 and positive integer powers of x:

Denition 2 Given constants a0, a1, , an, where an6= 0, the expression

P(x) = a0+ a1x+ a2x2+ · · · + anxn

is a polynomial of degree n The constant term is a0, and ak is the coecient of xkfor

k= 0, 1, n We will refer to a polynomial of degree less than or equal to n as a polynomial

of order n

20 CHAPTER 1 FUNCTIONS, LIMITS AND CONTINUITY

We may refer to the function dened by the polynomial P (x) simply as “the polynomial P (x)”

A polynomial of order 1 is a linear function, and a polynomial of degree 2 is a quadraticfunction The basic facts about linear and quadratic functions are reviewed in Section A5 ofAppendix A Polynomials of degree higher than 2 can be complicated In fact, the tools that

we will develop in calculus will help us understand the behavior of such functions

Example 5 Let

P(x) = 1 12x2+ 1

24x4.Thus, P (x) is a polynomial of degree 4 Figure 6 displays the graph of P ¤

1 2 3

Figure 6

Denition 3 A rational function is a quotient of polynomials Thus, f is a rational functionif

f(x) =p(x)q(x),where p(x) and q(x) are polynomials

Note that the natural domain of a rational function f = p/q consists of all x  R such that

q(x) 6= 0 Also note that negative-integer powers of x dene rational functions, since

xn= 1

xn.for each positive integer n

a) Express the domain of f as a union of intervals

b) Plot the graph of f with the help of your graphing utility

Solution

a) We have

x2Thus, the denominator of the expression for f (x) vanishes i and only if x = 1 or x = 1.Therefore, the domain of f is

(, 1)  (1, 1)  (1, +)

b) Figure 7 displays the graph of f , as produced by a graphing utility In Figure 6 the viewingwindow is [2, 2] × [10, 10] Since |F (x)| is very large if x is near 1 or 1 (you should samplesome values), the choice of an appropriate viewing window may require some experimentation.

We will discuss the behavior of a function such as f near points such as±1 in some detail inSection 1.6 ¤

Figure 7

The Functions Tangent and Secant

The trigonometric function tangent is the quotient of sine and cosine:

tan (x) = cos (x)sin (x)

at each x such that cos(x)6= 0 Since cos (x) = 0 if and only if x is an odd multiple of ±/2,i.e.,

x= ± (2n + 1)2where n = 0, 1, 2, , a real number is in the natural domain of the tangent function if and only

if it is not an odd multiple of±/2

The tangent function is periodic with period  Indeed,

= cos(x)sin(x) = tan(x)(we have made use of the addition formulas for sine and cosine, as indicated)

Since tangent is periodic with period , the graph of tangent on the interval (/2, /2) gives

an idea about the tangent function on the entire number line Figure 8 displays the graph oftangent on the interval (3/2, 3/2)

22 CHAPTER 1 FUNCTIONS, LIMITS AND CONTINUITY

The picture indicates that the graph of tangent is symmetric with respect to the origin Indeed,tangent is an odd function:

tan(x) = cos(x)sin(x) = sin(x)

cos(x) =  tan(x),since sine is odd and cosine is even

You can nd additional review material about tangent in Section A6 of Appendix A

Another special trigonometric function is secant that is the reciprocal of cosine Thus,

of the even function cosine:

sin(x).The trigonometric functions sine, cosine, tangent, and to a lesser extent secant, are importantspecial functions of calculus The functions cotangent and cosecant will not play a prominentrole

Composite Functions

Let’s begin with a specic case

Example 7 Let F (x) = sin(x2) If we set g (x) = x2 and f (x) = sin (x), we have F (x) =

f(g (x)) We can also view a function as an input-output mechanism, and consider the evaluation

of F (x) as a two-stage operation: Given the input x, the function g produces the output x2.This serves as the input of the sine function f and the result is the output sin¡

x2¢ We candisplay this view of F schematically as follows:

x xj 2 i sin(x2)

¤

Example 7 illustrates the operation of composing functions:

Denition 4 Given functions f and g, the function f g (read “f composed with g”) isdened so that

(f  g)(x) = f(g(x))for each x such that g (x) and f (g (x)) is dened

Thus, the domain of f g consists of all x in the domain of g such that g(x) is in the domain

of f The symbol “ ” is a little circle, and f g should not be confused with the product of

f and g A function that can be expressed as f g will be referred to as the composition of

f and g Such a function is a composite function We can describe the composite function

f g schematically:

x g(x)j f(g(x))iExample 8 Let f (x) = sin (x) and g (x) = x2, as in Example 7 Then

(f g) (x) = f (g (x)) = f¡x2¢

= sin¡

x2¢for each x R ¤

The order in which we compose functions matters, as illustrated by the following example:

Example 9 Let f (x) = sin(x) and g(x) = x2, as in Example 8 Show that

g f 6= f g

Solution

In Example 8 we determined that (f ... polynomial is periodic with period 2, and can be expressed as a linear bination of products of powers of sin(x) and cos(x) As we will discuss briey in Chapter 9,trigonometric polynomials provide useful...

so that f is a trigonometric polynomial, as a linear combination of sin (x) and sin (3x)

a) Conrm that f is periodic with period 2

b) Express f (x) as a linear combination of products...

appli-Denition A trigonometric polynomial is a linear combination of the constant function

1, and functions of the form sin(nx) and cos(nx), where n is an integer

A trigonometric polynomial